# CHAPTER 12

# SOLVING LINEAR EQUATIONS

# 12.1 Least-Squares Analysis

Consider a system of linear equations

where *A* ^{m×n}, *b* ^{m}, *m* ≥ *n*, and rank *A* = *n*. Note that the number of unknowns, *n*, is no larger than the number of equations, *m*. If *b* does not belong to the range of *A*, that is, if *b* ∉ (*A*), then this system of equations is said to be *inconsistent* or *overdetermined.* In this case there is no solution to the above set of equations. Our goal then is to find the vector (or vectors) *x* minimizing ||*Ax* − *b*||^{2}. This problem is a special case of the nonlinear least-squares problem discussed in Section 9.4.

Let *x** be a vector that minimizes ||*Ax* − *b*||^{2}; that is, for all *x* ^{n},

We refer to the vector *x** as ...